Chaotic vibrations of beams on nonlinear elastic foundations subjected to reciprocating loads

Chaotic vibration of beams resting on a foundation with nonlinear stiffness is investigated in this paper. Cosine–cosine function is employed in modeling of the reciprocating load. The equation of motion is derived and solved to obtain corresponding Poincaré section in phase–space. Lyapunov exponent as a criterion for chaos indication is obtained. Dynamic behavior of the beam is examined in resonance condition. Homoclinic orbits are captured and their corresponding Melnikov's functions are established. A parametric study is then carried out and effects of linear and nonlinear parameters on the chaotic behavior of the system are studied.     

Dynamic response of beams on generalised elastic foundations

Dynamic responses of beams on generalised elastic foundations is studied using the method of Initial parameters. The foundation model proposed by Vlasov and Leontev is modified by incorporating in the analysis the horizontal displacements in the elastic foundation thus making it more general and physically close to the actual situation. Results are compared with those reported by Rades, using Pasternak's foundation model and Winkler's model. The insufficiency of the Winkler's model in the study of dynamic responses (mainly the bending moments) is emphasized. Solutions presented are quite general for application to beams on generalised elastic foundations subjected to arbitrary external dynamic loads and (or) moments.     

Generalized solutions of beams with jump discontinuities on elastic foundations

Summary  The bending solutions of the Euler–Bernoulli and the Timoshenko beams with material and geometric discontinuities are developed in the space of generalized functions. Unlike the classical solutions of discontinuous beams, which are expressed in terms of multiple expressions that are valid in different regions of the beam, the generalized solutions are expressed in terms of a single expression on the entire domain. It is shown that the boundary-value problems describing the bending of beams with jump discontinuities on discontinuous elastic foundations have more compact forms in the space of generalized functions than they do in the space of classical functions. Also, fewer continuity conditions need to be satisfied if the problem is formulated in the space of generalized functions. It is demonstrated that using the theory of distributions (i.e. generalized functions) makes finding analytical solutions for this class of problems more efficient compared to the traditional methods, and, in some cases, the theory of distributions can lead to interesting qualitative results. Examples are presented to show the efficiency of using the theory of generalized functions. Received 6 June 2000; accepted for publication 24 October 2000     

Structural modeling of beams on elastic foundations with elasticity couplings

A new simplified structural model and its governing equations for beams on elastic foundations with elastic coupling are proposed. This modeling system is simple but appropriate for the initial structural design of large-scale submerged floating-beam structures moored by tension legs spaced at uniform interval along the beam. The model is actually for beam on discrete elastic supports rather than on continuous elastic foundations. Therefore, the governing equations are based on finite difference calculus and solutions for beams on discrete elastic supports with elasticity coupling are also proposed. To clarify the applicability limit of the proposed model, the equivalence between a beam on discrete elastic supports and that on continuous elastic foundation is investigated by comparisons of characteristic solutions.     

Winkler-Pasternak弹性地基梁自由振动的二维弹性分析

基于二维线弹性理论,应用Hamilton原理,获得Winkler-Pasternak弹性地基梁自由振动的控制微分方程,应用微分求积法(DQM)数值研究了梁自由振动的无量纲频率特性。计算结果与已有的结果(Bernoulli-Euler梁和Timoshenko梁)比较表明,本文的分析方法对弹性地基长梁和短梁自由振动的研究都有效。最后考虑了几何参数对梁频率的影响,以及不同边界条件下地基系数对频率的影响和收敛性。     

Vibrations and transient responses of Timoshenko beams resting on elastic foundations

Summary A finite element technique is used to obtain the natural frequencies and transient responses of Timoshenko beams resting on elastic foundations. The beam is discretized into beam elements, each with four degrees of freedom. The equations of motion in terms of the nodal degrees of freedom are derived by applying Hamilton's principle. The numerical results for a hinged-hinged beam are given to show the effects of rotatory inertia, shear deformation and foundation constants on the natural frequencies of the beam. The transient responses of beams are subsequently presented and compared with the available solutions.

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Eigenschwingungen und transientes Verhalten von Timoshenko-Balken mit elastischer Bettung

Übersicht Mit einer Finit-Element-Technik werden die Eigenfrequenzen und das transiente Verhalten von Timoshenko-Balken mit elastischer Bettung ermittelt. Der Balken wird dabei in Elemente mit je vier Freiheitsgraden diskretisiert. Die Bewegungsgleichungen werden mit Hilfe des Prinzips von Hamilton hergeleitet. Numerische Ergebnisse für einen beidseitig gelenkig gelagerten Balken zeigen den Einfluß von Drehträgheit, Schubverformung und Bettungskonstanten auf die Eigenfrequenzen. Anschließend wird das transiente Verhalten dargestellt und mit bekannten Ergebnissen verglichen.

     

CC series solution for bending of rectangular plates on elastic foundations

The analytical solution for the bending problem of the rectangular plates on an elastic foundation is investigated by using the Stockes' transformation of a double variables function. The numerical results for the rectangular plates with free edges on the elastic foundations under a concentrated force are given in the example. First Received Dec. 14 1993     

Effect of transverse shears on complex nonlinear vibrations of elastic beams

Models of geometrically nonlinear Euler-Bernoulli, Timoshenko, and Sheremet’ev-Pelekh beams under alternating transverse loading were constructed using the variational principle and the hypothesis method. The obtained differential equation systems were analyzed based on nonlinear dynamics and the qualitative theory of differential equations with using the finite difference method with the approximation O(h²) and the Bubnov-Galerkin finite element method. It is shown that for a relative thickness λ ⩽ 50, accounting for the rotation and bending of the beam normal leads to a significant change in the beam vibration modes.     

Nonlinear vibration of functionally graded graphene-reinforced composite laminated beams resting on elastic foundations in thermal environments

Modeling and nonlinear vibration analysis of graphene-reinforced composite (GRC) laminated beams resting on elastic foundations in thermal environments are presented. The graphene reinforcements are assumed to be aligned and are distributed either uniformly or functionally graded of piece-wise type along the thickness of the beam. The motion equations of the beams are based on a higher-order shear deformation beam theory and von Kármán strain displacement relationships. The beam–foundation interaction and thermal effects are also included. The temperature-dependent material properties of GRCs are estimated through a micromechanical model. A two-step perturbation approach is employed to determine the nonlinear-to-linear frequency ratios of GRC laminated beams. Detailed parametric studies are carried out to investigate the effects of material property gradient, temperature variation, stacking sequence as well as the foundation stiffness on the linear and nonlinear vibration characteristics of the GRC laminated beams.     

A certain approach to the solution of boundary value problems for the theory of plates and beams

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 25, No. 9, pp. 96–101, September, 1989.     

Shear stresses in elastic beams: an intrinsic approach

The theory of non-uniform flexure and torsion of Saint-Venant's beam with arbitrary multiply connected cross section is revisited in a coordinate-free form to provide a computationally convenient context. Numerical implementations, by Matlab, are performed to evaluate the maximum elastic shear stresses in beams with rectangular cross sections for different Poisson's ratios. The deviations between the maximum and mean stresses are then diagrammed to adjust the results provided by Jourawski's method.     

Forced nonlinear oscillations of semi-infinite cables and beams resting on a unilateral elastic substrate

In this work, we study the nonlinear oscillations of mechanical systems resting on a (unilateral) elastic substrate reacting in compression only. We consider both semi-infinite cables and semi-infinite beams, subject to a constant distributed load and to a harmonic displacement applied to the finite boundary. Due to the nonlinearity of the substrate, the problem falls in the realm of free-boundary problems, because the position of the points where the system detaches from the substrate, called Touch Down Points (TDP), is not known in advance. By an appropriate change of variables, the problem is transformed into a fixed-boundary problem, which is successively approached by a perturbative expansion method. In order to detect the main mechanical phenomenon, terms up to the second order have to be considered. Two different regimes have been identified in the behaviour of the system, one below (called subcritical) and one above (called supercritical) a certain critical excitation frequency. In the latter, energy is lost by radiation at infinity, while in the former this phenomenon does not occur and various resonances are observed instead; their number depends on the statical configuration around which the system performs nonlinear oscillations.     

Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation

In this study, simple analytical expressions are presented for large amplitude free vibration and post-buckling analysis of functionally graded beams rest on nonlinear elastic foundation subjected to axial force. Euler–Bernoulli assumptions together with Von Karman’s strain–displacement relation are employed to derive the governing partial differential equation of motion. Furthermore, the elastic foundation contains shearing layer and cubic nonlinearity. He’s variational method is employed to obtain the approximate closed form solution of the nonlinear governing equation. Comparison between results of the present work and those available in literature shows the accuracy of this method. Some new results for the nonlinear natural frequencies and buckling load of the FG beams such as the effect of vibration amplitude, elastic coefficients of foundation, axial force, and material inhomogenity are presented for future references.     

Variational approach to the solution of a contact problem for an elastic half-plane

Belorussian Polytechnic Institute, Minsk. Translated from Prikladnaya Mekhanika, Vol. 30, No. 7, pp. 70–73, July, 1994.     

Dynamic response of elastic plates on viscoelastic foundations to moving loads

Summary The steady-state response of an inifinite plate strip on a foundation of Kelvin material to a moving harmonic line load is presented. The line of application of the load is perpendicular to the infinite edges of the plate, and the load moves parallel to the edges at constant speed. The equations of motion for the plate are those of Mindlin, and they are solved by the integral transform technique. The uniqueness and existence of the physically reasonable solution for any speed of the moving load are discussed. The transient response of the same plate, but with semi-infinite length, is formulated as a mixed initial-boundary value problem. A numerical solution by the method of characteristics is proposed. The propagation of stress discontinuities along wave fronts is also investigated.

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Übersicht Es wird das Verhalten eines unendlich langen Plattenstreifens auf einem Fundament aus Kelvin-Material unter einer sich bewegenden, harmonisch über die Breite verteilten Linienlast nach Erreichung des Beharrungszustandes beschrieben. Die Angriffslinie der Last steht senkrecht auf den unendlich langen Kanten der Platte; die Last bewegt sich parallel zu den Kanten mit gleichförmiger Geschwindigkeit. Die Bewegungsgleichungen für die Platte sind dieselben wie die von Mindlin; sie werden durch eine Integral-Transformation gelöst. Die Eindeutigkeit und Existenz der physikalisch sinnvollen Lösung für beliebige Geschwindigkeiten der wandernden Last werden diskutiert. Der Einschwingvorgang derselben, von Null bis ins Unendliche erstreckten Platte wird als gemischtes Anfangs-Grenzwert-Problem behandelt. Eine numerische Lösung mit Hilfe des Charakteristiken-Verfahrens wird vorgeschlagen. Das Fortschreiten von Spannungssprüngen längs Wellenfronten wird ebenfalls untersucht.